First-principles and CALPHAD-type study of the Ir-Mo and Ir-W systems

This study presents the thermodynamic modeling of the Ir-Mo and Ir-W systems by means of the CALPHAD (CALculation of PHAse Diagrams) approach supported with the first-principles calculations. A critical evaluation of the phase equilibria and the thermodynamic property data in literature was conducted for both systems. Due to the lack of experimental data, the first-principles calculations were applied to obtain the enthalpies of the solid and intermetallic phases. The thermodynamic parameters were assessed using the PARROT module of Thermo-Calc. A set of self-consistent parameters for the Ir-Mo and Ir-W systems was obtained after the optimization. Satisfactory agreement between the calculated results and the experimental data, including phase equilibria and thermodynamic properties, has been achieved.


Introduction
Iridium is widely used in the high-temperature applications [1][2][3] including crucibles and car engine spark plugs owing to its high strength and excellent corrosion resistance [4,5]. However, the great brittleness [6,7] of iridium metal restricts its application. Alloying of other elements (such as Mo, W and Th) can not only significantly improve its processability [8,9] but also enhance its strength [10]. Moreover, the refractory metals (like Mo and W) with one of the most oxidation-resistant one (like Ir) would result in materials with particularly useful technological properties [11,12]. Therefore, it is worthy to obtain a fundamental knowledge of the Ir-Mo and Ir-W systems, such as phase equilibria and thermodynamic data, for designing appropriate compositions of iridium-based alloys. Although limited experimental data on phase equilibria of the Ir-Mo and Ir-W systems have been reported [13][14][15][16][17], no thermodynamic information is available in literature. First-principles calculations have been widely accepted as a reliable method of predicting thermodynamic data for CALPHAD modeling when experimental data are lacking or have large uncertainties [18], thus we use it to obtain the enthalpies of the solid and intermetallic phases in both systems. In this paper, we intend to integrate the first-principles calculations and CALPHAD [19] modeling to assess the Ir-Mo and Ir-W systems.

The Ir-Mo binary system
The Ir-Mo binary system consists of eight phases, including the Liquid, Fcc_A1, Bcc_A2 and five intermetallic compounds (IrMo3, IrMo, Ir3Mo, , ). The structural information of the phases is listed in Table 1. In 1954, Raub [20] investigated the Ir-Mo alloys at temperatures ranging from 800 ℃ to 1600 ℃ using X-ray diffraction (XRD), and observed the IrMo3 and  phases. Later on, Knapton [21] applied the same method to study the Ir-Mo alloys covering 15 to 85 at.% Ir and discovered the  phase. In 1963, Michalik and Brophy [15] constructed a phase diagram of the Ir-Mo system based on their experiments. Michalik and Brophy's samples were prepared with 99.9% Ir and 99.9% Mo powders and sintered in vacuum between 1200 ℃ and 1400 ℃ for an hour [15]. Only three intermetallic compounds (IrMo3,  and ) were detected [15] with metallography, XRD, fluorescence and electron microprobe analyses [15]. Five invariant reactions in this system were reported [15]: Liquid + Fcc_A1 ↔  at 2300 ℃, Bcc_A2 + Liquid ↔ IrMo3 at 2110±10 ℃, Liquid + IrMo3 ↔ at 2095±15 ℃, Liquid ↔at 2080±5 ℃ and ↔ IrMo3 at 1975±5 ℃. Furthermore, the solubility of Ir in Mo was determined as 16 at.% at 2110 ℃ and decreases to less than 5 at.% at 1500 ℃. The IrMo3 phase transforms peritectically from Bcc_A2 and Liquid, at around 2110 ℃. The composition of this phase varies from 23 at.% Ir at 2110 ℃ to a range of 22-25 at.% Ir at 1900 ℃. The  phase, is a stoichiometric compound, which transforms peritectically from the Liquid and IrMo3 phases at about 2095 ℃. In the report of Michalik and Brophy [15], the  phase can both appear in the eutectic and the eutectoid reactions, and it exists in a wide range from 37 at.% Ir at 2080 ℃ to 76 at.% Ir at 2300 ℃. The and Fcc_A1 two-phase region covers 76-78 at.% Ir at 2300 ℃. Besides, the lattice parameters of each phase were gained by their experiments [15], which are listed in Table 1. In 1965, Giessen [22] assessed the Ir-Mo binary system and proposed a new phase diagram. They added the IrMo and Ir3Mo phases in the diagram based on literature data and their own experiments [22]. IrMo is a stoichiometric compound indicated as the low-temperature phase of . Furthermore, the Ir3Mo phase was suggested to locate in the previous phase range [15]. Three more invariant reactions were included in the assessed phase diagram: Liquid + Fcc_A1 ↔ Ir3Mo, ↔ IrMo  Ir3Mo and ↔ IrMo IrMo3. The peritectic reaction Liquid + Fcc_A1 ↔  at 2300 ℃ summarized by Michalik and Brophy [15] was thus revised as Liquid+Fcc_A1 ↔ Ir3Mo due to the appearance of Ir3Mo. The invariant reactions are summarized in Table 2. Moreover, they measured the lattice parameters [22] of these two new phases using arcmelting, heat-treatments, metallographic and X-ray powder-diffraction technique, as listed in Table 1.

The Ir-W binary system
The Ir-W binary system consists of seven phases including the Liquid, Fcc_A1, Bcc_A2 and four intermetallic compounds (IrW, Ir3W, , ). The structural information of the phases is listed in Table 1. In 1951, Raub and Walter [13] firstly investigated the intermediate phases after the sample annealing at 1400-1800 ℃ and observed the  phase. The solidus and liquidus were studied by two research groups [14,16] using the incipient melting technique and the drop method, respectively. Rapperport and Smith [14] used 99.9% pure Ir and W while Tylkina et al. [16] used 99.8% pure Ir and "high purity W" (as written in literature) to prepare the samples by arc melting. In 1991, Nagender et al. [23] assessed this system and suggested the existence of the Ir3W and IrW phases above 1200 ℃ with hexagonal and orthorhombic crystal structures, respectively. Recently, Omori et al. [17] verified the existence of the Ir3W and IrW using the diffusion couple method and equilibrated alloys. In the latest assessed phase diagram [17], six invariant reactions exist: Liquid = + Fcc_A1, Liquid = + Bcc_A2 + Liquid = =Bcc_A2, Fcc_A1 + =r3W and =IrWBcc_A2. The eutectic reaction, Liquid = + Fcc_A1, appears at 2305 ℃ of 21 at.% W. The Fcc_A1 and two-phase region exists in the range of about 19~22 at.% W. Another eutectic reaction is Liquid = + appearing at 2460 ℃ of 70 at.% W. The content of W in the phase at 2460 ℃ is about 66 at.% [23]. The peritectic reaction, Bcc_A2 + Liquid = appears at 2540 ℃ of 75 at.% W and the content of W in the Liquid phase of this reaction is about 74 at.%. At 1810 ℃, the  phase transforms into the  and Bcc_A2 phases. The contents of W in the  and Bcc_A2 phases are 57 at.% and 96 at.% in this eutectoid reaction, respectively. The assessed maximum solubility of Ir in W is about 10 at.% [23]. All the invariant reactions are summarized in Table 2.

First-principles calculations
Formation enthalpies obtained by the first-principles calculations can be considered as "experimental data" [26][27][28][29] when such data are unavailable. In the present work, the first-principles calculations were conducted in the Vienna ab initio simulation package (VASP) [30] with generalized gradient approximation (GGA) [31] proposed by Perdew et al. [32] for the exchange and correlation contributions, and the valence electrons were treated by projector augmented plane-wave (PAW) potentials [33]. When the Hellman-Feynman forces were less than 10 -2 eV/Å, the atoms were deemed to be relaxed towards equilibrium. A plane-wave cutoff energy of 400 eV and an energy convergence criterion of 10 -5 eV for electronic structure self-consistency were adopted to make the total energy difference less than 0.1 kJ/mol-atoms. The method of performing Brillouin-zone integrations was Monkhorst-Pack k-point [34] generation mesh scheme. The unit cell size and the ionic coordinates were fully relaxed to find the stable state. After the final static calculation, the formation enthalpies of compounds and end-members at 0 K can be described as following: Here the total energy of a compound or end-member is E(IrmXn) (X = Ir, Mo, W). The total energies of reference states for pure elements are E(X). Lattice parameters and mixing enthalpies of Fcc_A1 and Bcc_A2 Ir-Mo and Ir-W solid solutions were investigated using first-principles calculations utilizing the special quasirandom structure (SQS) approaches [35,36]. 16 (5) is the binary interaction parameters and the coefficients of a and b are parameters to be optimized based on the available experimental data.

Intermetallic compounds
The intermetallic compounds of the Ir-Mo and Ir-W binary systems are all described by the two-sublattice models [41]. The occupation of atoms in this model can be written as, (Ir,M)x(Ir,M)y, where the subscripts of x and y represent the number of sites in each sublattice. The Gibbs free energy of the φ phase per mole-formula can be expressed by the following equation: : : : Where the parameter

Optimization
The thermodynamic model parameters of the Ir-Mo and Ir-W binary systems were evaluated on the basis of experimental information and assessed data using the PARROT module of Thermo-Calc software [42]. The step-by-step optimization procedure described by Du et al. [43] was utilized. Taking the Ir-Mo system as an example, the binary interaction parameters for the liquid phase, 0 Finally, all the parameters were evaluated together to provide the best description of the system. The assessed thermodynamic parameters are listed in Table 3. Table 3 Phase Thermodynamic parameters (J/mol)

The Ir-Mo binary system
The first-principles calculated lattice parameters of the pure Ir, Mo, and intermetallic compounds are shown in Table 1, in which the relative error range between the calculated value and literature data [15,22,24] is ±1.20%. The result shows that the error is acceptable. Fig. 1(a) shows the calculated lattice parameters of Fcc_A1 and Bcc_A2 by the SQS approach compared with the experimental data [15]. The relative error range of this phases is ±1.03%. The calculated atomic volumes compared with the experimental data [15] is shown in Fig. 1(b) and the relative error range is ±2.29%. All the calculated results agree well with the experimental ones [15]. Fig. 2(a) shows the mixing enthalpies of Fcc_A1 and Bcc_A2 in the Ir-Mo binary system with the SQS approach and the CALPHAD method. The optimized mixing enthalpies accord well with the SQS models. The formation enthalpies of the Ir-Mo phases are presented in Fig. 2(b), in which the scattered points are the data obtained by the first-principles calculations. The CALPHAD results differ ±4.72kJ/mol-atoms from the first-principles data, which is reasonable agreement. The optimized phase diagram of Ir-Mo is shown in Fig. 3(a). The maximum solubility of Ir in (Mo) and Mo in (Ir) was calculated to be 15.56 at.% at 2116 ℃ and 21.84 at.% at 2293 ℃, respectively. The compositions of the and IrMo3 phases are very close, and thus an enlarged diagram is presented in Fig.  3(b). Fig. 3(c) exhibits the calculated phase diagram compared with the experimental data [15]. The measured data of the phase boundary between Fcc_A1 and Ir3Mo have an error of 25 ℃, which have a large uncertainty [15]. Therefore, these data were set a lower weight in the optimization. Nine invariant reactions were calculated, including, Liquid + Fcc_A1 = Ir3Mo, Liquid = + Ir3Mo, Bcc_A2 + Liquid = IrMo3, Liquid + IrMo3 = Liquid = = IrMo3, =IrMo  Ir3Mo, =IrMo IrMo3 and IrMo3 = IrMo + Bcc_A2. In Fig. 4(a), the peritectic reaction Liquid + Ir3Mo = assessed byGiessen et al. [22]which appears in the high-temperature range above 2100 ℃ is replaced by a eutectic reaction Liquid = + Ir3Mo in the present work. The calculated reaction can be regarded as tentative because no experimental information can verify the reaction type [22]. Moreover, they assumed the existence of two eutectoid reactions, i.e., =IrMo  Ir3Mo and =IrMo IrMo3, analogy to the Mo-Pt system [44]The temperatures of these eutectoid transformations were predicted to be 1491 ℃ and 1477 ℃, respectively. In table 2, a summary of the invariant equilibria of the binary Ir-Mo system is given to compare with the experimental and optimized temperatures and compositions.   The comparison of the lattice parameters between the calculated results and experimental data of Ir-W is shown in Table 1, the relative error range of which is ±2.63%, which is a good agreement. Fig. 4(a) shows the mixing enthalpies of Fcc_A1 and Bcc_A2 calculated by the SQS approach and CALPHAD method. The formation enthalpies of the compounds calculated by the first-principles calculations can be seen in Fig. 4(b), which were used as initial values for the optimization. The optimized mixing enthalpies have a good agreement with the SQS models. Fig. 5(a) shows the optimized phase diagram of Ir-W, which includes seven invariant reactions: Liquid = + Fcc_A1, Liquid = + Bcc_A2 + Liquid = =Bcc_A2, Fcc_A1 + =r3W, =IrWBcc_A2 and = IrW + Ir3W. In the present work, the peritectic reaction, Bcc_A2 + Liquid =  was given a lower weight than the eutectoid reaction at 1810 ℃ due to the large uncertainty of the experimental data. The comparison of invariant equilibria between the experimental data [14,16,17,20] and calculated results are listed in Table 2. Further comparison between the experimental data [14,16,17,20] and calculated results is shown in Fig. 5(b). It is indicated that the experimental phase equilibria data can be reproduced by the present CALPHAD-type calculations.

Conclusions
(1) The lattice parameters calculated by the first-principles calculations accord well with the experimental data, which demonstrates that the calculations are reliable. Formation enthalpies of the compounds and end-members calculated by the firstprinciples calculations were used as input during the CALPHAD-type optimization. (2) A set of self-consistent thermodynamic parameters of the Ir-Mo and Ir-W systems have been obtained by the CALPHAD approach. All the reliable experimental information can be satisfactorily reproduced by the present optimized parameters. Acknowledgments This work was financially supported from the National Natural Science Foundation of China (51171046, 51701232), Natural Science Foundation of Fujian Province (2018J01754), Key Laboratory of Eco-materials Advanced Technology (Fuzhou University), Fujian Province University (STHJ-KF1708). K. Chang acknowledges the CAS Pioneer Hundred Talents Program. Fig. 1. Lattice parameters of the Ir-Mo system: (a) the obtained results of Bcc_A2 by the first-principles calculations compared with the experimental data [15]; (b) the calculated atomic volumes compared with the experimental data [15]. Fig. 2. Thermodynamic data of the Ir-Mo system: (a) mixing enthalpies of Fcc_A1 and Bcc_A2 calculated by the CALPHAD (at 298 K) and SQS (at 0 K) approaches; (b) formation enthalpies obtained by the CALPHAD (at 298 K) and the first-principles (at 0 K) calculations.    [13,14,16,17]. Table 1 Crystal structure data of the phases in the Ir-Mo and Ir-W systems. Table 2 Experimental invariant reactions compared with the calculated results of the Ir-Mo and Ir-W systems. Table 3 Summary of the thermodynamic parameters in the Ir-Mo and Ir-W binary systems optimized in the present work.